I noticed that there is a structural similarity between the syntactic operations of Bealer's logic (see my paper "Bealer's Intensional Logic" that I uploaded to Researchgate for my interpretation of these operations) and the notion of non-symmetric operad. However for the correspondence to be complete I need a diagonalisation operation.
Consider an operad P with P(n) the set of functions from the cartesian product X^n to X.
Then I need operations Dij : P(n) -> P(n-1) which identify variables xi and xj.
Has this been considered in the literature ?
The idea of diagonalization in operad theory has been studied in the literature, although it is typically formulated in terms of "partial compositions" rather than "variable identification" as in your proposed Dij operation.
One approach to diagonalization in operads is to define a "partial composition" operation that takes two elements of an operad P and produces a new element by composing them along a diagonal. More precisely, given elements f ∈ P(m) and g ∈ P(n), we define their diagonal composition f ∘g ∈ P(m+n-1) as follows:
(f ∘g)(x1,...,xm+n-1) = f(x1,...,xi,...,xm,g(i-m+1),...,g(n))
where i is the unique index such that i-m+1 ≤ j < i for all j ∈ {1,...,m+n-1}.
This partial composition operation satisfies some important algebraic properties and has been extensively studied in the context of operad theory. However, it may not be directly applicable to your specific problem of identifying variables in an operad.
Another approach to diagonalization in operad theory involves the use of "modular operads", which are operads that allow for the composition of operations in a non-symmetric fashion. Modular operads provide a powerful framework for studying algebraic structures that arise in geometry, topology, and mathematical physics, and they have been used to study a wide range of phenomena, including Feynman diagrams, string field theory, and knot invariants.
In summary, while the specific operation you propose (Dij : P(n) -> P(n-1)) may not have been studied in the literature, there are related concepts in operad theory that may be useful for your purposes, such as diagonal composition and modular operads. I would recommend exploring these ideas further to see if they can be adapted to your specific problem.
Agnieszka Matylda Schlichtinger
It seems that there is a nice geometric way to visualise the Bealer operations similar to braids used in some areas of category theory (or the work of Bob Coecke et al. on quantum protocols). It is however somewhat more complex. To apply the "composition" operation you need to untwist or unravel parts of the associated graph first. I have not gotten around to working this out yet, but it seems to be a worthwhile reasearch project !
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